What M1 Does
An assay result below the lower limit of quantification (LLOQ) is not an exact concentration of zero. It indicates that the concentration was not quantified and that the result lies below the assay's stated threshold. M1 is a simple way to handle such records: remove each BQL observation record from the estimation data and fit the model to the quantified observations that remain.
The deletion applies to an observation, not automatically to an entire individual. If an individual has quantified records as well as BQL records, the quantified records remain available for estimation. If every outcome record for an individual is BQL, that individual contributes no concentration likelihood under M1. A subject record or covariate row may still be retained for data-management reasons, but it does not create outcome information for the concentration model.
This is different from recording a BQL result as zero or replacing it with a numerical concentration. M1 supplies no replacement value: it removes the BQL row from the estimation dataset. The original laboratory record should nevertheless be preserved outside that analysis dataset so that the deletion is transparent and can be audited.
From Laboratory Records to the M1 Dataset
Suppose \(y_{ij}\) is the measured concentration for individual \(i\) at observation time \(t_{ij}\), and let \(L_{ij}\) be the applicable LLOQ. A quantified record supplies a concentration value. A BQL record supplies the information \(y_{ij}<L_{ij}\), but M1 discards that record for estimation. The assay limit can differ across records; the deletion rule applies using the limit that belongs to each result.
Worked Example: Creating an M1 Analysis Dataset
Consider a hypothetical individual with a single-dose concentration profile. The assay LLOQ is \(0.10\) mg/L at every listed time, and the laboratory has classified the results shown below. The recorded BQL results are below the applicable LLOQ; no exact concentration is available for those rows.
| Time (h) | Recorded result (mg/L) | Status | Use in M1 estimation? |
|---|---|---|---|
| 0.5 | 0.82 | Quantified | Yes |
| 1 | 0.61 | Quantified | Yes |
| 2 | 0.39 | Quantified | Yes |
| 4 | 0.21 | Quantified | Yes |
| 8 | 0.11 | Quantified | Yes |
| 12 | Below 0.10 | BQL | No |
| 24 | Below 0.10 | BQL | No |
The M1 estimation data for this individual contain five concentration rows: \(0.82,\ 0.61,\ 0.39,\ 0.21,\) and \(0.11\) mg/L at \(0.5,\ 1,\ 2,\ 4,\) and \(8\) h. The rows at 12 and 24 h are omitted; they are not converted to zero or to \(0.10\) mg/L. This dataset shows a key consequence of M1: the quantified observations define the late observed profile, while the two later records provide no contribution to estimation.
In practice, the analysis dataset should keep each retained result paired with the correct individual, time, dose history, and other model-required fields. A record-level deletion must not accidentally delete that individual's dosing record or other quantified outcomes.
Likelihood After Deletion
Let \(f(y_{ij}\mid \eta_i,\theta)\) denote the model's conditional density for a quantified observation, given individual random effects \(\eta_i\) and population parameters \(\theta\). Under M1, define \(Q_i\) as the set of quantified observation indices for individual \(i\). The conditional likelihood uses only those records:
For a population model with random effects distributed according to \(g(\eta_i\mid\theta)\), the individual likelihood contribution is obtained by integrating over those random effects:
A BQL record is absent from the product. In particular, M1 does not include the probability that an observation fell below its LLOQ. If \(Q_i\) is empty, the product over quantified records is an empty product, equal to 1; integrating the random-effects density then gives 1. Thus, in a concentration-only analysis, an individual with no quantified concentrations contributes no concentration likelihood under M1.
Worked Example: The Likelihood Contribution M1 Omits
Use a hypothetical single observation with an additive normal residual model. At one time, the model predicts a mean concentration of \(0.08\) mg/L, the residual standard deviation is \(0.05\) mg/L, and the LLOQ is \(0.10\) mg/L. Assume the BQL classification is correctly recorded and the conditional residual distribution is normal.
The probability of a BQL result under this model is
A likelihood that uses the BQL status would include this probability as a contribution. M1 deletes the record instead, so its contribution is 1 on the likelihood scale, or 0 on the log-likelihood scale. The omitted log-likelihood contribution is \(\log(0.6554)\approx -0.4225\). This numerical comparison is illustrative; M1 does not calculate the probability and then subtract it. It simply does not use the BQL row.
For contrast, suppose a different quantified observation is \(0.45\) mg/L, with a model-predicted mean of \(0.40\) mg/L and additive normal residual standard deviation \(0.10\) mg/L. Its density contribution is
The corresponding log-density is approximately \(1.258\). M1 retains this quantified result and its density contribution, while omitting the BQL result and its event-probability contribution. The assumptions for these calculations are the stated normal conditional error model and the given means and standard deviations; the calculations do not assert that this residual model is suitable for every concentration analysis.
Why Terminal-Phase Deletion Matters
A BQL record still carries information: it says the measured concentration fell below a known threshold. When BQL records are concentrated in the terminal phase, deletion removes precisely the observations that could help distinguish how quickly concentrations decline near and below the quantification limit. The quantified data may still support an estimate of elimination, but the terminal observations provide no direct constraint under M1.
The consequence depends on the sampling design and model. For a declining one-compartment profile, for example, the predicted concentration may be written \(C(t)=C_0e^{-kt}\), where \(C_0\) is the initial level and \(k\) is the elimination rate constant. At late times, a BQL status can constrain the predicted concentration relative to the LLOQ and therefore inform \(k\). Deleting that status discards the constraint. It does not prove that the fitted \(k\) is wrong, but it leaves less information for estimating it and may make the estimate more dependent on earlier quantified observations and model assumptions.
A short justification makes the information loss explicit. For a BQL observation with normal conditional distribution \(Y\sim N(\mu,\sigma^2)\), the probability of falling below a fixed LLOQ \(L\) is \(\Phi(z)\), where \(z=(L-\mu)/\sigma\). Its log-likelihood contribution is \(\log\Phi(z)\). Since \(dz/d\mu=-1/\sigma\), differentiation gives
For finite \(z\), this derivative is generally nonzero: the BQL event can change the likelihood's preference for values of \(\mu\). If \(\mu\) depends on a parameter such as \(k\), the event can also inform that parameter through the chain rule. M1 removes this contribution. This establishes why deleting a BQL result is not equivalent to keeping all the information in the laboratory record.
Assumptions, Diagnostics, and Failure Modes
M1 is an estimation-data rule, not a model that explains why results are BQL. Its use does not make deletion harmless. It is most defensible as a pragmatic approximation when few records are BQL, those records do not carry important information for the parameters of interest, and enough quantified observations remain to identify the model. These are considerations to assess, not guarantees that M1 is unbiased.
In many concentration profiles, the chance of being BQL is related to the underlying concentration, which itself depends on time and model parameters. Thus, BQL status is not ordinarily equivalent to an unrelated missing value. A core limitation is that M1 ignores this outcome-dependent pattern. If many subjects have no quantified terminal observations, or if the BQL runs cover much of the terminal phase, estimates and uncertainty measures based only on retained concentrations do not account for the discarded event information.
- Count and locate the deletions. Tabulate BQL and quantified records by time, individual, dose occasion, and assay limit. A large concentration of deletions in the terminal phase is a warning that M1 removes a structured part of the profile.
- Check retained terminal coverage. Confirm how many individuals have quantified samples late enough to inform the terminal decline. If late quantified data are sparse, the fitted terminal behavior may be sensitive to which records remain.
- Check for all-BQL individuals. Identify individuals with no retained concentration records. Under M1 they add no concentration outcome contribution; verify that the estimation software has not treated a placeholder, zero, or stale value as an observation.
- Audit data transformations. Confirm that BQL rows were excluded rather than coded as zero, that quantified values near the LLOQ were retained when classified as quantified, and that each row used the correct assay limit and time.
- Review fit and precision diagnostics. Examine whether parameter uncertainty, terminal predictions, or fit stability are sensitive to the reduced data. A numerically successful fit does not show that discarded censoring information was unimportant.
The assumptions behind the worked dataset example were that its assay classifications and LLOQ were correct and the listed quantified concentrations were the data supplied to estimation. The likelihood calculation additionally assumed the specified conditional normal residual distribution. In an applied analysis, those details need verification; M1 cannot correct a wrong BQL flag, an incorrectly assigned LLOQ, or an unsuitable model for quantified observations.
How M1 Differs From Other Handling Principles
M1 makes no numerical claim about a BQL concentration. A substitution approach instead inserts a chosen value for each BQL row, thereby treating an unquantified result as if it were an exact concentration for estimation. A censored-likelihood approach retains the event information by using the probability of an observation falling below its applicable LLOQ. These are different data and likelihood choices, not alternative names for deletion.
The appropriate approach depends on the scientific aim, the amount and pattern of censoring, the residual-error model, and the reliability of the assay information. M1 is simple to implement and can be transparent when deletions are limited. Its simplicity does not restore the discarded information, and a comparison of fit quality alone cannot establish that the deletion rule is adequate. Any analysis using M1 should report how many records were removed, where they occurred, and which records remained for estimation.
Common Mistake
Wrong reasoning: “A BQL result is effectively zero, so deleting it has no effect on the analysis.”
Correction: BQL means the result was below a specified threshold, not that the concentration was zero. M1 discards even that threshold information. A terminal BQL result can constrain the predicted profile, so removing it can affect the information available for parameter estimation even though no exact concentration was measured.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- An individual has four quantified concentrations followed by three BQL records. Under M1, which records contribute to the concentration likelihood?
- A dataset has an LLOQ of \(0.05\) mg/L. A result is reported as BQL. What concentration should be entered for that record in an M1 estimation dataset?
- Why can deleting terminal-phase BQL observations matter for an elimination-rate estimate?
- Under M1, what is the concentration-likelihood contribution of an individual whose every concentration record is BQL?
- Name two checks that help determine whether M1 has removed a potentially important part of the data.
Solutions. (1) The four quantified records contribute; the three BQL records are omitted. (2) No concentration is entered for that record in the M1 estimation dataset; the BQL row is excluded, not set to zero or \(0.05\) mg/L. (3) A late BQL status indicates that concentration fell below the assay threshold and can constrain the declining profile. Deleting it removes that constraint, which may leave less information about the elimination rate. (4) It contributes no concentration outcome information under M1; the product over its quantified records is empty. (5) Examples include tabulating BQL records by time and individual, checking how many quantified terminal observations remain, and identifying individuals with no quantified records.